stability of cylindrical transonic shocks for two

STABILITY OF CYLINDRICAL TRANSONIC SHOCKS FOR

TWO–DIMENSIONAL STEADY COMPRESSIBLE EULER SYSTEM

LI LIU AND HAIRONG YUAN

Abstract. For given supersonic flow passing a divergent nozzle, if the pressure at the exit of the

nozzle (back pressure) is sufficiently large, then a transonic shock may appear in the nozzle. The

pressure increases across the shock front, while the supersonic flow jumps down to subsonic. This

paper is devoted to analyze such phenomena by establishing the stability of a class of cylindrical

symmetric transonic shocks for two–dimensional complete compressible steady Euler system. This

result also partly explains the effectiveness of the popular quasi–one–dimensional model of nozzle

flows used in aerodynamics. Mathematically, this is to solve a nonlinear free boundary problem

for an elliptic–hyperbolic composite system, with the circular transonic shock front as the free

boundary. We accomplish this by finding the (locally) unique fixed point of an appropriately defined

boundary profile updating mapping. To define this mapping, we encounter a series of nonclassical

boundary value problems on an annulus, which involve a new type of nonlocal elliptic problem, and

integral–like solvability conditions to determine the position of the free boundary. This reflects an

interesting new feature of boundary value problems of elliptic–hyperbolic composite systems.

1. Introduction

This paper is devoted to establish the stability of steady cylindrical symmetric transonic shocks

for the two–dimensional complete compressible Euler system. It will help us understand those

transonic shocks appearing in the divergent part of de Laval nozzles, as well as the effectiveness of

the popular quasi–one–dimensional model of nozzle flows.

In 1903, Stoola firstly observed in experiments the following fascinating phenomena [1, 10, 23,

24, 29]: for a given subsonic flow which accelerates in the convergent part and becomes to be

supersonic after passing the throat (the narrowest part of the nozzle), if the pressure at the exit of

the nozzle (back pressure) lies in a certain interval, then a surface of discontinuity may appear in

the divergent part of the nozzle. The pressure increases across this surface, while the supersonic

flow ahead of this surface jumps down immediately to subsonic flow. The position, profile and

strength of the discontinuity can be automatically adjusted by the back pressure. Such a surface of

discontinuity is called a transonic shock front, to distinguish from those shock fronts for which both

the flow states ahead and behind of them are supersonic (for example, those weak shocks attached

to a slim wedge in supersonic flows [10, 29]).

By the development of computational techniques, such transonic shock phenomena had also been

demonstrated by various numerical simulations of nozzle flows, and conversely, it is also widely used

to test algorithms, schemes designed for capturing shocks [1]. Due to their numerous important

applications in practice, such as the design of inlets of certain types of engines of supersonic jets [29]

and supersonic wind tunnels [23], there have a tremendous amount of works devoted to understand

2000 Mathematics Subject Classification. 35F15, 35J25, 35J55, 35M20, 76H05, 76N10.

Key words and phrases. compressible Euler system, stability, transonic shocks, free boundary problem, hyperbolic-

elliptic composite system, nonlocal elliptic problem.

1

2 LI LIU AND HAIRONG YUAN

the transonic shocks by proposing various different models and analyzing these models with either

theoretical or computational or experimental methods. For subsonic–supersonic flows in nozzles

and many interesting numerical and experimental results, one may consult [1, 9, 17, 22, 24, 29].

For the quasi–one–dimensional model [1, 10, 24, 29], in steady flow case, Courant and Friedrichs

explained the transonic shock phenomena by several algebraic functions (see section 145 of [10]).

For the non-steady flow case, in a series of papers ([20, 21]), Liu showed that the supersonic and

subsonic flows are dynamically stable, and for transonic flows, the shock waves tend to decelerate

along a divergent nozzle and accelerate along a convergent nozzle. Thus the transonic shocks in

divergent nozzles are dynamically stable. These results rely on the Glimm theory of one–space–

dimensional hyperbolic balance laws [11]. It might be surprising that, although the quasi–one–

dimensional model is rather restrictive and idealized at first glance, the derived results conform to

the experiments very well (see section 8.1 of [24]). As in [21], the result established in the present

paper may also partly clarify this wonder.

Further understanding of the transonic shocks requires progresses in the theory of elliptic bound-

ary value problems in non-smooth domains and new ideas to treat free boundaries — the transonic

shock fronts. In recent years, there appear many outstanding works in this direction, for example,

[2, 3, 4, 5, 6, 7, 13, 25]. Yuan [27] established the ill–posedness of a class of transonic shock problems

in a two–dimensional slowly–varying duct for given pressure at the exit. It is shown that the back

pressure can only be given with a constant difference to make the problem well–posed; that is, the

back pressure should contain a number to be determined by the upstream supersonic flow and the

profile of the duct. Similar result has also been obtained for three–dimensional complete steady

Euler system for flows in straight ducts with constant square sections under certain symmetric

assumptions on the coming supersonic flows at the entrance by Chen and Yuan [8].

But such ill–posedness results contradict the observed transonic shock phenomena, since ill–

posedness indicates instability of transonic shocks, hence it is not likely to be observed in nature.

Why this happens?

The reason is that the above works [2, 3, 4, 5, 6, 8, 25, 26, 27] depend on a special transonic

shock solution (the first class of transonic shocks as called in [28]), for which the nozzle is a straight

duct, the transonic shock front is flat, and both the states ahead and behind of the shock front are

uniform. Thus the position of the shock front may be arbitrary, and the back pressure does not

depend on the position of the shock front (see [28]).

Thus to study the transonic shock phenomena, one should turn to other special solutions. We

notify that in section 145 of [10], Courant and Friedrichs showed that in some cases the quasi–one–

dimensional model is identical to the spherical symmetric solutions of the potential flow equations.

In another words, the quasi–one–dimensional flow coincides with the spherical symmetric flow.

Since the quasi–one–dimensional model really works to some extent in understanding the transonic

shock phenomena, the special spherical symmetric transonic shock solutions of steady Euler system

(called as the second class of transonic shocks in [28] ) might be a proper point for starting rigorous

mathematical analysis. The paper [28] is devoted to the construction of such special solutions by

carefully analyzing the reduced ordinary differential equations and Rankine–Hugoniot conditions.

The analogue of spherical symmetric flows in two–dimensional case is called cylindrical symmetric

flows: they only depend on the distance from a fixed point. We note that a special case of Theorem

STABILITY OF TRANSONIC SHOCKS 3

6.1 in [21] by Liu implies that for unsteady cylindrical symmetric flows, the second class of transonic

shocks are dynamically stable, i.e., stable with respect to small perturbations of initial datum. The

purpose of this paper is to establish the stability of the second class of transonic shocks in two–

dimensional case with respect to the perturbations of upcoming supersonic flows. The main result

may be roughly stated as the following theorem. (See Theorem 3.2 and 3.3 for a precise version.)

Theorem. For steady Euler flows, in polar coordinates, if the supersonic flow at the entrance

(r0, θ) : θ ∈ [0, 2π] is nearly cylindrical symmetric, then there exists an interval such that if the

pressure at the exit (r1, θ) : θ ∈ [0, 2π] (r1 > r0 > 0) lies in this interval, there will appear uniquely

one nearly cylindrical symmetric transonic shock in the annulus (r, θ) : r0 < r < r1, 0 ≤ θ ≤ 2π.

Combining Liu’s theorem and the present result, we see that the cylindrical symmetric transonic

shocks are stable in a very strong sense. This fact may partly explain why we can observe the

transonic shock phenomena, and why the quasi–one–dimensional model works surprisingly well.

Now we describe in a rough way the procedure to prove this result and comment on several dif-

ficulties encountered in establishing it. Comparing to [27], there are some interesting new features.

The flow between the entrance and the transonic shock front is supersonic. To determine it,

since the steady Euler system for supersonic flow is hyperbolic, one solves a periodic initial value

problem for a (symmetric) hyperbolic system. General results in this direction have been obtained

by many scholars ([18]). So we just present the existence and estimate of supersonic flow in the

annulus in section 3 and omit the detailed proof.

Obtaining the transonic shocks requires to solve a free boundary problem (FB). The transonic

shock front is the free boundary, which is a curve needs to be solved simultaneously with the

subsonic flow behind it. For convenience, we define the position and profile of the free boundary as

follows.

Definition. Let Σ : ξ = ψ(η), η ∈ [0, η0] be a free boundary. We call r = ψ(0) the position of Σ,

and the curve ξ = ψ(η) − r the profile of Σ.

Thus we can determine a free boundary once we know its position and profile. The point to

introduce the profile and the position of a transonic shock front is that they are determined by

different mechanisms. The profile is solved by parts of the Rankine–Hugoniot conditions, while the

position depends on the integral–like solvability conditions of elliptic boundary value problems. In

[2, 3, 4, 5, 6, 25, 26, 27], one fixes the position of the transonic shock front artificially.

The problem (FB) is solved by showing that a boundary profile updating mapping (BPUM) has

uniquely one fixed point via the well known Banach contraction mapping principle (BCMP): Any

contractive mapping on a complete metric space has one and only one fixed point. This mapping

maps a candidate profile of the free boundary Υ to a new one Υ and is formulated as a Cauchy

problem for an ordinary differential equation, which originates in the separating of the Rankine–

Hugoniot conditions. See section 3.7.

However, to define the BPUM properly one requires to solve a “semi-fixed” nonlinear boundary

value problem of a nonlocal elliptic system, which is called problem (C(Υ)). Here, by recalling that

we have fixed the profile of the candidate free boundary as Υ, “semi-fixed” means that the position

of the candidate free boundary needs to be solved simultaneously with the solution. For example,

an elliptic problem in a rectangle [r, 1] × [0, 1] is “semi-fixed” if the number r is also unknown.


The reason why the position can be determined is that the linearized nonlocal elliptic problem

is of negative Fredholm index; that is, it is uniquely solvable if and only if certain integral–like

conditions hold.

We explain now why nonlocal elliptic problems rise here. As exposition of [27], the combination

of Lagrangian transformation and characteristic decomposition is a powerful tool to treat the two–

dimensional complete compressible steady Euler system. For subsonic flow, this elliptic–hyperbolic

composite system can be reduced to a first order elliptic system on the plane and two algebraic

equations: the Bernoulli’s law and invariance of entropy for smooth flows along streamlines. This

is why we encounter here purely elliptic problems. This approach avoids loss of derivatives, thus

permits us to use the standard simplified Newton’s method to linearize the nonlinear problems and

solve them by BCMP.

However, due to the special structure of the second class of transonic shocks, there rise first

order terms of the candidate free boundary in the Taylor expansion of the nonlinear boundary

conditions on the candidate free boundary, which are another part of the separated Rankine–

Hugoniot conditions (see section 4.1). By combining the BPUM to cancel the explicit dependence

on the candidate free boundary in the boundary conditions, we are led to nonlocal elliptic problems

involving integrals. After careful reformulations, we obtain a class of boundary value problems for

second order nonlocal elliptic equations (see, for example, (5.13)). To our best knowledge, such

problems have not been proposed and studied before.

For these linear nonlocal elliptic problems, we first establish Fredholm alternative in H1(Ω) by

the classical Lax–Milgram theorem, where Ω is a square with periodical conditions on the upper

and lower boundaries. Then we show uniqueness by separation of variables (i.e., completeness of

Fourier series in L2 for periodic functions), thus we get existence in H1. Then we can write the

nonlocal terms as nonhomogeneous terms, and use W 2,2 as well as Schauder estimates for equi-

valued surface problems, Robin problems of second order uniformly elliptic equations to improve

the regularity of these solutions [14, 15, 19].

The problem (C(Υ)) is solved by applying the BCMP to a nonlinear mapping defined by the

above studied linearized problem of (C(Υ)) in a careful way (see section 6). To determine the

position, the definition of the mapping is more complex than [27].

There are also several technical aspects. For example, since for different Υ, the problem (C(Υ))

is defined in different domains, so to obtain contraction, we used some homeomorphisms (4.32) at

appropriate times to normalize the domains to a square. Some auxiliary functions also need to be

introduced carefully to simplify the resultant nonlocal elliptic problems. We study the stability of

transonic shocks in an annulus instead of a two–dimensional divergent nozzle to avoid the difficulty

of singularities caused by the walls of the nozzles, but at a price to take care to assure the transonic

shock is periodical; that involves using cancellation techniques and solving equi-valued surface

boundary value problems.

The rest of the paper is organized as follows. In the preliminary section 2, we use Lagrangian

transformation and characteristic decomposition to write the two–dimensional steady complete

Euler system in polar coordinates as a 2 × 2 system coup;ed with two algebraic equations. The

annulus and boundary conditions are also illustrated. Section 3 formulates the transonic shock

problem rigorously and states the main result of this paper. By assuming the solvability of problem


(C(Υ)), we also complete a proof of the main result. From section 4, we need only focus on problem

(C(Υ)). Section 4 devotes to reformulate this problem in an equivalent, but more transparent form.

Section 5 studies the solvability and estimates of the corresponding linearized problems. The main

ideas are, by subtracting the background solution (i.e., the second class of transonic shocks) and

using Taylor expansions to pick out the first order terms and higher order terms, and introducing

wisely some “potentials” to write the nonlocal boundary value problem of a first order elliptic

system as two boundary value problems of nonlocal elliptic equations of second order. Finally, in

section 6, by constructing a nonlinear mapping, we solve (C(Υ)) by applying BCMP.

2. Lagrangian Formulation of Euler System in Polar Coordinates and

Characteristic Decomposition

2.1. Euler System in Polar Coordinates. We consider in this paper polytropic gas, i.e., perfect

fluids with the constitutive relation p = A(S)ργ . Here p, ρ, S are the pressure, density, entropy of

the flow respectively, and γ ∈ (1,∞) is the adiabatic exponent. So the local speed of sound is

a =√

∂p/∂ρ =√

γA(S)ργ−1 =√

γp/ρ.

In the polar coordinates (r, θ), let u, v respectively be the velocity components along r, θ direc-

tions, then the two–dimensional steady Euler system may be written as:

• Divergence form in polar coordinates

∂r(rρu) + ∂θ(ρv) = 0, (2.1)

∂r(rρu2 + rp) + ∂θ(ρuv) − (ρv2 + p) = 0, (2.2)

∂r(rρuv) + ∂θ(ρv2 + p) + ρuv = 0, (2.3)

1

2(u2 + v2) +

a2

γ − 1= c0. (2.4)

• Symmetric form in polar coordinates

ρu 0 1

0 ρu 0

1 0 uρa2

∂r

u

v

p

+

1

r

ρv 0 0

0 ρv 1

0 1 vρa2

∂θ

u

v

p

+1

r

−ρv2

ρuv

u

= 0,

1

2(u2 + v2) +

a2

γ − 1= c0.

(2.4) is the well known Bernoulli’s law holding along streamlines with the constant c0 which may

be different on different streamlines.

In the polar coordinates, let an annulus be

N := (r, θ) : 0 < r0 ≤ r ≤ r1, 0 ≤ θ ≤ 2π. (2.5)

We denote the entrance (r, θ) : r = r0, 0 ≤ θ ≤ 2π as Σ0, and the exit (r, θ) : r = r1, 0 ≤ θ ≤ 2πas Σ1.


2.2. Euler System in Lagrangian Coordinates. By similar methods as in [27], due to (2.1),

we may introduce the Lagrangian coordinates (ξ, η) with r = ξ, θ = θ(ξ, η) to simplify the structure

of the Euler system. Here η represents the mass flux between two streamlines. Following similar

computations as in [27], by introducing a new variable w = v/u, we obtain the Euler system in

Lagrangian coordinates as:

• Divergence form in Lagrangian coordinates

∂ξ

(

1

ξρu

)

− ∂η

(

w

ξ

)

= 0, (2.6)

∂ξ

(

ξu+ξp

ρu

)

− ∂η (ξpw) − ρ(u2 + v2) + p

ρu= 0, (2.7)

∂ξv + ∂ηp+v

ξ= 0, (2.8)

1

2(u2 + v2) +

a2

γ − 1= c0. (2.9)

• Symmetric form in Lagrangian coordinates

u 0 1ρ

0 u 01ρ 0 u

(ρa)2

∂ξ

u

v

p

+

0 0 −v0 0 u

−v u 0

∂η

u

v

p

+1

ξ

−v2

uvuρ

= 0,

1

2(u2 + v2) +

a2

γ − 1= c0. (2.10)

By applying the characteristic decomposition method as in [6, 7, 27], we can write the first

equation in (2.10) as

∂ξw +(1 −M2)a2

u(a2 − u2)∂ηp−

(u2 + v2)

(a2 − u2)ξw + λR∂ηw = 0,

∂ξp−ρ2a2u3

a2 − u2∂ηw +

ρa2(u2 + v2)

(u2 − a2)ξ+ λR∂ηp = 0,

(2.11)

and

∂ξ

(

p

ργ

)

= 0, (2.12)

i.e., invariance of entropy for C1 flows along streamlines. Where M :=√u2 + v2/a is the Mach

number, and λR = ρva2/(u2 − a2). As showed in [27], (2.11)(2.12) together with Bernoulli’s law

are equivalent to (2.10) provided ρu 6= 0, u 6= a,M 6= 1, in spite of the flow is supersonic (M > 1)

or subsonic (M < 1). The system (2.11) is elliptic for subsonic flow, and hyperbolic for supersonic

flow.

Since (2.12) can be integrated once its value on the shock front is known, and the constant c0 in

Bernoulli’s law (2.9) can be determined from the supersonic flow at the entrance of the annulus,

thus by characteristic decomposition we can write the Euler system as a 2×2 system (2.11) coupled

with two algebraic equations.

The advantage of Lagrangian transformation is that it straightens the streamlines. Now N

becomes NL := (ξ, η) : r0 ≤ ξ ≤ r1, 0 ≤ η ≤ η0, with the total mass flux in the annulus

η0 =∫ 2π0 r0ρu(r0, s)ds being determined by the supersonic coming flow at the entrance. The upper

boundary Γ1 := (ξ, η) ∈ NL : η = η0 and the lower boundary Γ0 := (ξ, η) ∈ NL : η = 0 both


correspond to the same streamline passing the point (r0, 0) in polar coordinates. So, the boundary

conditions on them should be periodical with the period η0.

Now suppose the equation of the shock front in NL is ξ = ψ(η), η ∈ [0, η0], then if [p] 6= 0, by

similar manipulation as in [6, 7, 27], the Rankine–Hugoniot conditions on shock front corresponding

to (2.6)–(2.8) can be written as

ψ′(η) = [uw]/[p], (2.13)

G1(U(ψ(η), η), U−(ψ(η), η)) := [uw][w] + [p]

[

1

ρu

]

= 0, (2.14)

G2(U(ψ(η), η), U−(ψ(η), η)) := [uw][pw] + [p]

[

u+p

ρu

]

= 0. (2.15)

Here U−(ψ(η), η) is the limit value of the supersonic flow ahead of the shock front, and U(ψ(η), η)

is the limit value of the subsonic flow behind of the shock front, and [·] as usual denotes the jump

of a quantity across the shock front [10].

3. Formulation of Problems and Main Results

In this section we present the transonic shock problem (TS) in NL rigorously, together with

some closely connected sub-problems. The main results are also precisely stated.

3.1. Transonic Shock Problem (TS) in NL. This is the following boundary value problem in

Lagrangian coordinates:

(TS) :

(2.11)(2.12)(2.9) in NL,

U = U(r0)(η)

:= (u(r0)(η), v(r0)(η), p(r0)(η), ρ(r0)(η)) on ξ = r0,

Periodical conditions on η = η0, η = 0,

p = p(r1) on ξ = r1.

(3.1)

Here ξ = r0, ξ = r1 are respectively the entrance and exit of NL, and U = (u, v, p, ρ)t. U(r0)(η)

is a given supersonic state at the entrance, with u(r0)(η) > 0, ρ(r0)(η) > 0. p(r1) is a given

appropriately large pressure at the exit. Periodical conditions on η = 0 and η = η0 mean that all

the functions involved are periodical with respect to η with the period η0. Obviously the constant

c0 in Bernoulli’s law (2.10) is now

c0 =1

2(u(r0)

2 + v(r0)2) +

γ

γ − 1

p(r0)

ρ(r0).

Generally speaking, c0 is a function of η. Later on, for simplicity of computations and presentation,

we assume that c0 is the same on every streamline. This assumption is not restrictive also from

physical viewpoint, since for example, for de Laval nozzles, if the flow is uniform at the entrance,

then the Bernoulli’s constant does not depend on streamlines.

One may write out the problem (TS) in Euler formulation (for instance, Descartes coordinates

or polar coordinates) according to the Lagrangian transformation employed before. One may also

translate all the results stated below in Lagrangian formulation into Euler formulation. Since these

are routine and note that by Lemma 4.6 in [3] (or Proposition 2.4 in [27]), the estimates will not

be affected, so we omit them here.


3.2. Background Solution. In [28] Yuan has established a class of special solutions to problem

(TS), by assuming that U(r0) and p(r1) are constants in (3.1), and the flow is cylindrical symmetric,

i.e., depends only on ξ. For convenience, we call problem (TS) in this simplified case as problem

(TSbg), which is exactly problem (TSP) in [28]. We summarize the results in [28] (Corollary 8) as

the following theorem.

Theorem 3.1 (Special solutions of (TSbg)). For given supersonic state U(r0) at ξ = r0 with

u(r0) > 0, there exists 0 < pmin < pmax such that for p(r1) ∈ [pmin, pmax], there exists a unique

transonic shock solution (U−, U+; rs) of (TSbg). Here ξ = rs ∈ [r0, r1] is the transonic shock

front separating the supersonic flow U− in [r0, rs] and the subsonic flow U+ in [rs, r1] with entropy

condition p+(rs) > p−(rs) holding.

Definition 3.1 (Background solution). In the following we call the above solutions (U−, U+; rs) of

(TSbg) as a background solution of problem (TS) determined by a constant supersonic state U(r0)

and a back pressure p(r1), and was denoted as (U−b , U

+b ; Σb) with Σb : ξ = rs being the shock front.

The purpose of this article is to show the stability of the background solution under perturbations

of the coming supersonic flow: if U(r0)(η)−U−b (r0) is small in some norm, then there also exists a

unique transonic shock front ξ = ψ(η), with a supersonic flow U− ahead of it, a subsonic flow U+

behind it, and U− − U−b , U+ − U+

b , ψ − rs can be controlled by U(r0)(η) −U−b (r0) in some norm.

Since the background solution satisfies the quasi–one–dimensional model, this stability result,

together with that of [21], may partly explain why the quasi–one–dimensional model conforms to

the experiments in a surprising fashion ([24], section 8.1).

So it is important to get familiar with the background solutions. We have the following remarks.

Remark 3.1. (1) Theorem 6 in [28] claims that: for any rs ∈ [r0, r1], U−b (r0) and rs determine

uniquely one p+b (r1). It decreases strictly as rs increases. We denote this function as p+

b (r1)(rs).

Furthermore, by continuous dependence on initial datum of solutions of ordinary differential equa-

tions, one sees that p+b (r1)(rs) is uniformly Lipschitz continuous with respect to rs.

(2) The supersonic state U−b can be defined in the whole annulus by using supersonic curves (see

Proposition 1 in [28]).

(3) There exists a positive constant h0 determined by U−b (r0) such that U+

b can be defined by

left extension of subsonic curves in [rs − h0, r1] (see Proposition 9 in [28]). Later we will always

use background solution in the understanding of such extensions. In addition, it is natural that

U+b (ξ, η) = U+

b (ξ), since background solution does not depend on η.

3.3. Main Results. Let the equation of the shock front be ξ = ψ(η), η ∈ [0, η0]. Set

Ωψ :=

(ξ, η) : η ∈ (0, η0), ψ(η) < ξ < r1

, (3.2)

and denote the upper and lower wall as

Γ1 :=

ξ ∈ [ψ(η0), r1], η = η0

, Γ0 :=

ξ ∈ [ψ(0), r1], η = 0

.

This paper is devoted to prove the following theorem:

Theorem 3.2. Let Ub :=(

U−b , U

+b ; Σb := ξ = rs

)

be a background solution of problem (TS).

There exist positive constants ε0 and C0 depending solely on Ub such that if the following assump-

tions hold in problem (TS):


(a) U(r0)(η) is a small perturbation of U−b (r0): it is a periodic vector-valued function of η with

period η0 and∥

∥U(r0) − U−b (r0)

∥

∥

C5([0,η0];R4)≤ ε ≤ ε0; (3.3)

(b) For background solution, p+b (r1) is appropriately taken such that rs ∈ (r0 + C0ε, r1 − C0ε)

holds. In addition, in (3.1) we take

p(r1) = p+b (r1), (3.4)

then (TS) has a unique solution (U−, U+; Σ) which satisfies the following:

(1) U− is supersonic, U+ is subsonic, and Σ is the transonic shock front separating U−, U+

and satisfies entropy condition; in addition, they are all periodic with respect to η with the

period η0;

Let the equation of Σ be ξ = ψ(η) and define the subsonic region Ωψ as (3.2), supersonic

region as

N−L :=

(ξ, η) ∈ NL, ξ < ψ(η)

. (3.5)

(2) (U−, U+; Σ) is a small perturbation of the background solution, that is, for any α ∈ (0, 1),

the following estimates hold:

∥

∥U− − U−b

∥

∥

3+α;N−

L

< C0ε, (3.6)∥

∥U+ − U+b

∥

∥

2+α;Ωψ< C0ε, (3.7)

‖ψ − rs‖3+α;[0,η0] < C0ε. (3.8)

Here ‖·‖k+α;Ω := ‖·‖Ck,α(Ω).

Combining this result with Theorem 3.1, we have the following conclusion:

Theorem 3.3. Under the assumptions of Theorem 3.2, there exists an interval [pεmin, pεmax] ⊂

[pmin, pmax] such that if in (3.1) p(r1) ∈ [pεmin, pεmax], then problem (TS) has a unique transonic

shock solution, which is a small perturbation of the background solution determined by U−b (r0) and

p(r1).

3.4. Perturbed Supersonic Flow. Our first goal is to obtain the unique existence of the super-

sonic flow in NL subjected to the following problem (SF), by using theory of semi-global classical

solutions of quasi-linear hyperbolic systems with periodical boundary conditions:

(SF) :

(2.11) in NL,

w = w(r0)(η), p = p(r0)(η) on ξ = r0,

Periodical conditions on Γ0,Γ1,

where p,w are unknowns in (2.11). ρ appeared in the coefficients is determined by

ρ(ξ, η) = ρ(r0)(η)

(

p(ξ, η)

p(r0)(η)

) 1γ

,

while by Bernoulli’s law, u is solved from

ρ =γ

γ − 1· p

c0 − 12u

2(1 + w2).


Lemma 3.1. Under the assumptions of Theorem 3.2, there exists an ε0 > 0 depending only on

U−b (r0) such that (SF) exists uniquely one solution U−, which is supersonic, periodical with respect

to η with period η0 and the following estimate holds:

∥

∥U− − U−b

∥

∥

C4(NL;R4)≤ C0ε, (3.9)

where C0 relies solely on U−b (r0).

Proof. The proof may be based on those methods introduced, for example, in chapter 1 of [18]. We

omit it here.

3.5. A Free Boundary Value Problem. By Lemma 3.1, to prove Theorem 3.2, we need only

to solve the following free boundary problem (FB), which determines the transonic shock front

simultaneously with the subsonic flow behind it.

Problem (FB): Find the functions ξ = ψ(η) and U such that for U− obtained in

Lemma 3.1, there holds

(2.11)(2.12)(2.9) in NL,

(2.13) − (2.15) on ξ = ψ(η),

Periodical conditions on η = η0, η = 0,

p = p+b (r1) on ξ = r1.

Our purpose is to demonstrate the following theorem.

Theorem 3.4. There exist constants ε0, C0 depending only on U−b (r0) and rs such that under the

assumptions of Theorem 3.2, problem (FB) has a unique solution (U,ψ) which is periodical with

respect to η with period η0 and satisfies:

∥

∥U − U+b

∥

∥

2+α;Ωψ< C0ε, ‖ψ − rs‖3+α;[0,η0] < C0ε. (3.10)

Proof of Theorem 3.2. By combination of Lemma 3.1 and Theorem 3.4, Theorem 3.2 is obvious.

3.6. “Semi-Fixed” Boundary Value Problems (C). Problem (FB) may be regarded as solving

a fixed point of a boundary profile updating mapping. Construction of such a mapping involves series

of “semi-fixed” boundary value problems of Euler system. Here “semi-fixed” means that although

the geometric shape of part of the boundary is known, its position still needs to be determined.

We set

Sσ : =

ψ∗(η) ∈ C3,α[0, η0] : ψ∗ is periodical with period η0,

‖ψ∗‖3+α;[0,η0]≤ σ, ψ∗(0) = 0

, (3.11)

where σ ≤ σ0 < h0/2 with h0 the number occurring in Remark 3.1 (3). Later in section 6 we also

need to introduce

Pκ :=

r ∈ R1 : |r − rs| ≤ κ ≤ κ0 <

h0

2

. (3.12)

Here σ0, κ0 are small constants to be chosen later. Roughly speaking, Sσ is the set of those possible

curves representing the profile of the shock front, and Pκ is the set of possible positions of the shock

front (i.e., it passes the point with Lagrangian coordinates (r, 0)).


Now for any given ψ∗ ∈ Sσ, we consider the following “semi-fixed” boundary problem (C(ψ∗))

(or (C) for simplicity). This is to solve r∗ and U , such that by setting

ψ(η) := ψ∗(η) + r∗, (3.13)

then in Ωψ (see (3.2)) the equations (2.11)(2.12)(2.9) hold, and the periodical boundary conditions

on Γ0, Γ1 are true, as well as p = p+b (r1) on the exit. In addition, on the candidate shock front

Σψ :=

ξ = ψ(η)

, we require that

G1(U(ψ(η), η), U−(ψ(η), η)) = 0, (3.14)

G2(U(ψ(η), η), U−(ψ(η), η)) = F2(U(ψ(η), η), U−(ψ(η), η), ψ(η)), (3.15)∫ η0

0

[uw]

[p]

∣

∣

∣

∣

U(ψ(η),η),U−(ψ(η),η)

ds = 0. (3.16)

Here G1, G2 are defined as in (2.14)(2.15), and

F2(U(ψ(η), η), U−(ψ(η), η), ψ(η))

= µ0

(

ψ∗(η) −∫ η

0

[uw]

[p]

∣

∣

∣

∣

(U(ψ(s),s),U−(ψ(s),s))

ds

)

with µ0 the following positive constant:

µ0 :=1

rs

(

1

ρ+b (rs)u

+b (rs)

)

(p+b (rs) − p−b (rs))

2. (3.17)

We see from above that Ωψ, the definition domain of the problem (C(ψ∗)) is to be determined,

while only up to freedom one — that is, r∗ = ψ(0) is unknown. These problems are neither totally

free boundary problems like (FB), nor fixed boundary problems. So we call them “semi-fixed”

boundary problems. Solvability of such problems originates from elliptic boundary value problem

with negative Fredholm index as to be shown later by analysis of linearized problems. The reader

may also wonder the appearance of the above annoying term F2. It really brings us lots of trouble

later in analysis of linearized problem, but is necessary to get a convergent iteration scheme (BCMP)

to solve the nonlinear problem. We will explain this further in next section. The utility of (3.16)

will be clear in the next subsection.

For problem (C), we have the following theorem which most of the rest of this paper is attributed

to prove.

Theorem 3.5 (Unique existence of solution of (C(ψ∗)) and its continuous dependence on ψ∗). There

exist constants ε0,M1, C1 relying solely on Ub such that under the assumptions of Theorem 3.2, for

any given ψ∗ ∈ Sσ with σ = M1ε, ε < ε0, problem (C(ψ∗)) has a unique solution (r∗, U∗) which is

periodical with respect to η with period η0 and satisfies the following estimates:

|r∗ − rs| ≤ C1ε,∥

∥U∗ − U+b

∥

∥

2+α;Ωψ< C1ε. (3.18)

Here ψ is defined by (3.13), and Ωψ by (3.2).

Furthermore, for j = 1, 2 and ψ∗j ∈ Sσ, let the solution of problem C(ψ∗

j ) be (r∗j , U∗j ), ψj = ψ∗

j+r∗j .

Then the following estimate holds:

|r∗1 − r∗2| +∥

∥(U∗1 − U+

b )(ψ1(η), η) − (U∗2 − U+

b )(ψ2(η), η)∥

∥

2+α;[0,η0]

≤ C1ε ‖ψ∗1(η) − ψ∗

2(η)‖3+α;[0,η0] . (3.19)


Remark 3.2. The choice of M1 and ε0 can only be determined later in studying of boundary profile

updating mapping. Their solely dependence on background solution is explained in Remark 6.1 in

section 6.

As shown in the next subsection, this theorem is the milestone of proof of Theorem 3.4.

3.7. Boundary Profile Updating Mapping. For any ψ∗ ∈ Sσ, problem C(ψ∗) has uniquely a

solution (r∗, U∗). By setting ψ = ψ∗ + r∗ as in (3.13), then the following expression, which based

on equation (2.13) that deduced from Rankine–Hugoniot conditions:

ψ∗(η) =

∫ η

0

[uw]

[p]

∣

∣

∣

∣

(U∗(ψ(η),η),U−(ψ(η),η))

ds (3.20)

established a mapping Λ : ψ∗ 7→ ψ∗. We call it a boundary profile updating mapping.

We observe that, once the above boundary profile updating mapping has a fixed point ψ∗, then

F2, the right hand side of (3.15), vanishes. Thus all (2.13)–(2.15), i.e., the Rankine–Hugoniot

conditions, hold along the curve ξ = ψ∗(η) + r∗ — so it is exactly the desired shock front in

problem (FB). Further, U∗ is the subsonic state behind it. Here (r∗, U∗) is the solution of problem

C(ψ∗).

We will use Banach contraction mapping principle (BCMP) to show Λ has a fixed point. As-

suming Theorem 3.5 is true for the moment, we can prove the following lemma.

Lemma 3.2. There exist constants ε0, C2 depending only on Ub such that under the assumptions

of Theorem 3.2, Λ is a contractive mapping on Sσ with σ = C2ε. That is, there holds

∥

∥ψ∗∥∥

3+α;[0,η0]≤ C2ε. (3.21)

Proof. 1. By (3.16), and the periodic property of ψ∗ and U∗, we see that ψ∗ is a periodic function

with period η0. By estimates (3.9)(3.18) and the fact that [p] 6= 0 for background solution (i.e.,

physical entropy condition [10]), we see (3.21) holds. We may choose M1 in Theorem 3.5 to be C2

here, so Λ is an inner mapping on Sσ.2. Now using the notations in Theorem 3.5, let Λ(ψ∗

j ) = ψ∗j . Then by the estimate (3.9), and

mean value theorem, we get

∥

∥(U− − U−b )(ψ1, η) − (U− − U−

b )(ψ2, η)∥

∥

2+α;[0,η0]

≤ C4ε ‖ψ∗1(η) − ψ∗

2(η)‖3+α;[0,η0] ,

thus by applying estimates (3.18) (3.19),

∥

∥ψ∗1(η) − ψ∗

2(η)∥

∥

3+α;[0,η0]

≤ C5

∥

∥(U∗1 − U+

b )(ψ1(η), η) − (U∗2 − U+

b )(ψ2(η), η)∥

∥

2+α;[0,η0]

+C5ε ‖ψ∗1(η) − ψ∗

2(η)‖3+α;[0,η0] ≤ C6ε ‖ψ∗1(η) − ψ∗

2(η)‖3+α;[0,η0] .

The constant C6 depends only on Ub. By choosing ε0 further small such that C6ε0 < 1, then Λ

contracts.

Proof of Theorem 3.4. Lemma 3.2 and BCMP show that Λ has a unique fixed point satisfying

(3.21), so the discussion earlier in this subsection demonstrates Theorem 3.4 to be true under the

assumption that Theorem 3.5 is true.


By far our solely duty is to prove Theorem 3.5. We begin with rewriting problem (C) in an

equivalent, but more transparent and tractable form.

4. An Equivalent Form of Semi-Fixed Boundary Problem

The main idea involved in this section is to separate first order terms versus high order terms in

the equations and boundary conditions of problem (C) in the spirits of simplified Newton’s method.

This is necessary to linearize the original nonlinear problem. Although some computations are

tedious, the results are somewhat surprising since nonlocal elliptic problems arise.

4.1. Reformulation of Boundary Conditions. The purpose of this subsection is to rewrite

(3.14)(3.15) and (3.16).

1. Recall the definition of F2 above (3.17). Let F1 = 0. Then both (3.14)(3.15) take the form

G = F . Note that for background solution we have G(U+b (rs, η), U

−b (rs, η)) = 0, therefor G = F is

equivalent to the following expressions:

∂+G(U+b (rs, η), U

−b (rs, η)) • (U(ψ(η), η) − U+

b (ψ(η), η)) (4.1)

= −(

∂+G(U+b (ψ(η), η), U−

b (ψ(η), η)) − ∂+G(U+b (rs, η), U

−b (rs, η))

)

•(U(ψ(η), η) − U+b (ψ(η), η))

+

∂+G(U+b (ψ(η), η), U−

b (ψ(η), η)) • (U(ψ(η), η) − U+b (ψ(η), η))

−(

G(U(ψ(η), η), U−(ψ(η), η)) −G(U+b (ψ(η), η), U−(ψ(η), η))

)

−

G(U+b (ψ(η), η), U−(ψ(η), η)) −G(U+

b (ψ(η), η), U−b (ψ(η), η))

−

G(U+b (ψ(η), η), U−

b (ψ(η), η)) −G(U+b (rs, η), U

−b (rs, η))

+F.

We use “•” here as the scalar product of vectors, and ∂+G(U,U−) (∂−G(U,U−)) as the gradient of

G(U,U−) with respect to the variables U(U−), and U is (u,w, p)t, with w = v/u. Note that then

ρ can be expressed by Bernoulli’s law (2.9).

For simplicity, we set g1j , g

2j , g

3j respectively as the terms in the first three brackets “” of (4.1)

with j = 1, 2, that corresponds to Gj . All these are high order terms, as can be shown by differential

mean value theorem (Taylor expansions). On the contrary, by the same theorem, those in the forth

bracket of (4.1) involves first order terms of ψ(η) − rs:

G(U+b (ψ(η), η), U−

b (ψ(η), η)) −G(U+b (rs, η), U

−b (rs, η))

=(

∂+G(U+b (rs, η), U

−b (rs, η)) •

dU+b

dξ(rs)

+∂−G(U+b (rs, η), U

−b (rs, η)) •

dU−b

dξ(rs)

)

·(ψ(η) − rs) +O(ψ(η) − rs)2. (4.2)

Here O(g) = O(1)g, and the Landau symbol O(1) denotes a quantity who satisfies a uniform bound,

depending only on the background solution.


2. Now let us calculate explicitly for G1, G2. As in [27] we may get

∂±G(U+b (rs, η), U

−b (rs, η))

=∂(G1, G2)(U+, U−)

∂(u±, w±, p±)

∣

∣

∣

∣

(U+b

(rs),U−

b(rs))

= ±(

a±1 0 b±1a±2 0 0

)

,

where

a±1 = −2c0 + u±b (rs)2

2c0 − u±b (rs)21

ρ±b (rs)(u±b (rs))2

[p] < 0,

b±1 = −[p]1

ρ±b (rs)u±b (rs)p

±b (rs)

< 0, a±2 =[p]

γ

(

1 − (a±b (rs))2

(u±b (rs))2

)

< 0,

and [p] = p+b (rs) − p−b (rs) > 0.

For the first “()” in the right hand side of (4.2), by the equations satisfied by background solution

[28], we may get it is zero if G = G1. However, if G = G2, then it equals

µ0 :=[p]2

rs

1

ρ+b (rs)u

+b (rs)

> 0.

So by definition of F2, one obtains that for

µ1 =u+b (rs)

p+b (rs) − p−b (rs)

> 0,

there holds

F2 −

G2(U+b (ψ(η), η), U−

b (ψ(η), η)) −G2(U+b (rs, η), U

−b (rs, η))

+O(ψ(η) − rs)2

= µ0(rs − r∗) − µ0µ1

∫ η

0w(ψ(s), s) ds

+µ0

(

µ1

∫ η

0w(ψ(s), s) ds−

∫ η

0

[uw]

[p]

∣

∣

∣

∣

(U(ψ(s),s),U−(ψ(s),s))

ds)

+O(ψ(η) − rs)2. (4.3)

Let the last two terms in the above identity be g42 , and

g41 := F1 −

G1(U+b (ψ(η), η), U−

b (ψ(η), η)) −G1(U+b (rs, η), U

−b (rs, η))

.

Then g41 = O(ψ(η) − rs)

2. Both g41 , g

42 are higher order terms. We see here clearly that F is

introduced to cancel the first order term in (4.2), and further to make the boundary conditions do

not depend explicitly on the boundary. This is one of the reasons why nonlocal elliptic problems

occur later.

Therefor by straightforward calculations (4.1) turns out to be

(p(ψ(η), η) − p+b (ψ(η), η)) + µ2

∫ η

0w(ψ(s), s) ds

= g1 + µ3(r∗ − rs), (4.4)

(u(ψ(η), η) − u+b (ψ(η), η)) + µ5

∫ η

0w(ψ(s), s) ds

= g2 + µ4(r∗ − rs). (4.5)


Here

µ2 = −a+1 µ0µ1

a+2 b

+1

> 0, µ3 =a+

1 µ0

a+2 b

+1

< 0, (4.6)

µ4 = −µ0

a+2

> 0, µ5 =µ0µ1

a+2

< 0, (4.7)

g1 =1

b+1

4∑

k=1

gk1 − a+1

a+2 b

+1

4∑

k=1

gk2 , g2 =1

a+2

4∑

k=1

gk2 . (4.8)

(4.4)(4.5) are our desired form of conditions on the semi-fixed boundary Σψ. Note that g1, g2 are

higher order terms and µj (j = 1, · · · , 5) are nonzero constants.

3. As in (4.3), (3.16) may be written equivalently as∫ η0

0w(ψ(s), s) ds = h(U(ψ(η), η), U−(ψ(η), η)), (4.9)

with

h(U(ψ(η), η), U−(ψ(η), η))

=1

µ1

∫ η0

0

(

µ1 −u+b (ψ(s))

p+b (ψ(s)) − p−b (ψ(s))

)

w(ψ(s), s) ds

+1

µ1

∫ η0

0

(

u+b (ψ(s))

p+b (ψ(s)) − p−b (ψ(s))

− u(ψ(s), s)

p(ψ(s), s) − p−(ψ(s), s)

)

w(ψ(s), s) ds

+1

µ1

∫ η0

0

u−(ψ(s), s)

p(ψ(s), s) − p−(ψ(s), s)w−(ψ(s), s) ds := h1 + h2 + h3. (4.10)

For i = 1, 2, 3, hi are all consist of higher order terms.

4.2. The Perturbed Equations. Set

w1 = w, w2 = p− p+b ,

the purpose of this subsection is to write out the equations satisfied by w1, w2 in an elegant form.

We begin with (2.11).

1. Let

e1 = e1(ξ) :=(1 −M2)a2

u(a2 − u2)

∣

∣

∣

∣

U+b

> 0, d1 = d1(ξ) :=u2

(u2 − a2)ξ

∣

∣

∣

∣

U+b

, (4.11)

D1 = D1(ξ, η) = exp(

∫ ξ

ψ(η)d1(s)ds

)

. (4.12)

Then the first equation in (2.11) can be written as

∂ξ(D1w1) + ∂η(D1e1w2) = f1, (4.13)

with

f1 = D1(ξ, η)

−λR∂ηw +v2w

(a2 − u2)ξ+(

e1 −(1 −M2)a2

u(a2 − u2)

)

∂ηp

+(

d1 −u2

(u2 − a2)ξ

)

w

−D1(ξ, η)d1(ψ(η))ψ′(η)e1w2, (4.14)

which consists of higher order terms.


2. Let

H = H(U) =ρa2u2

ξ(u2 − a2)=γp

ξ

u2

u2 − a2.

It is crucial to note here that the relation of dependence is as follows:

u2 =2(c0 − a2

γ−1)

1 + w2, a2 =

γp

ρ;

according to the invariance of entropy along streamlines (2.12),

ρ = ρ(ξ, η) = ρ(ψ(η), η)

(

p(ξ, η)

p(ψ(η), η)

)1γ

.

Then due to Bernoulli’s law,

ρ(ψ(η), η) =γ

γ − 1

p(ψ(η), η)

c0 − 12u(ψ(η), η)2(1 + w(ψ(η), η)2)

.

Therefore

H(U) = H(p(ξ, η), w(ξ, η), p(ψ(η), η), u(ψ(η), η), w(ψ(η), η)), (4.15)

which depends particularly on the values of p, u,w on the boundary Σψ. This phenomenon reflects

the transportation effect of the “hyperbolic part” on the “elliptic part” of the Euler system. This

is also another reason why we are led to nonlocal elliptic problems.

By setting

e2 = e2(ξ) :=ρ2a2u3

a2 − u2

∣

∣

∣

∣

U+b

> 0, (4.16)

and using the fact that p+b (ξ) satisfies the ordinary differential equation [28]

dp

dξ=

−ρu2a2

ξ(u2 − a2),

the second equation in (2.11) may be written as

∂ξw2 − ∂η(e2w1) + (H(U) −H(U+b ))

= −λR∂ηw2 +ρa2v2

(a2 − u2)ξ+( ρ2a2u3

a2 − u2− e2

)

∂ηw. (4.17)

The three terms in the right hand side of the above equation are all higher order terms. We need

further to extract first order terms in H(U) −H(U+b ) by Taylor expansions.

3. Taylor expansions of H(U) − H(U+b ). For convenience, we use ∂jH(U)(j = 1, · · · , 5) to

denote the partial derivative of H with respect to the j-th variable in (4.15) at U . We denote

that U = (p(ξ, η), w(ξ, η), p(ψ(η), η), u(ψ(η), η), w(ψ(η), η)) in this paragraph for simplicity. So for

example, U+b = (p+

b (ξ, η), 0, p+b (ψ(η), η), u+

b (ψ(η), η), 0).

Note that H(U) is an analytical function of U , then by Taylor expansions up to second order

and (4.4)(4.5), one has

H(U) −H(U+b ) =

5∑

j=1

∂H(U+b ) • (U − U+

b ) +O(|U − U+b |2)

= d2(ξ)w2 + g3(ξ, η) + h(ξ)h1(ψ(η))(r∗ − rs)

+h(ξ)h2(ψ(η))

∫ η

0w(ψ(s), s) ds +O(|U − U+

b |2). (4.18)


Here we have set

d2(ξ) :=a+b (ξ)4(2 −M+

b (ξ)2 + γM+b (ξ)4)

ξ(u+b (ξ)2 − a+

b (ξ)2)2> 0, (4.19)

g3(ξ, η) := −γ − 1

γ· 2c0(a

+b (ξ))4

ξ(u+b (ξ)2 − a+

b (ξ)2)2· ρ+

b (ξ)

p+b (ψ(η))

·(

g1 + ρ+b (ψ(η))u+

b (ψ(η))g2)

, (4.20)

h(ξ) := −γ − 1

γ· 2c0(a

+b (ξ))4

ξ(u+b (ξ)2 − a+

b (ξ)2)2· ρ+

b (ξ) < 0, (4.21)

h1(ψ(η)) := p+b (ψ(η))−1

(

µ3 + µ4ρ+b (ψ(η))u+

b (ψ(η)))

, (4.22)

h2(ψ(η)) := −p+b (ψ(η))−1

(

µ2 + µ5ρ+b (ψ(η))u+

b (ψ(η)))

. (4.23)

It is easy to check that h1(rs) < 0, h2(rs) = −u+b (rs)/rsp

+b (rs) < 0, thus h2(ψ(η)) is bounded away

from zero if |ψ(η) − rs| is small.

4. Substituting (4.18) into the equation (4.17), we get

∂ξw2 − ∂η(e2w1) + d2(ξ)w2 + h(ξ)h1(ψ(η))(r∗ − rs) (4.24)

+h(ξ)h2(ψ(η))

∫ η

0w(ψ(s), s) ds

= −λR∂ηw2 +ρa2v2

(a2 − u2)ξ+( ρ2a2u3

a2 − u2− e2

)

∂ηw − g3(ξ, η) −O(|U − U+b |2).

The right hand side of the above equation, denoted as “♠” below, consists of higher order terms.

Now define

d3(ξ) := exp(

∫ ξ

r1

d2(s) ds)

> 0, (4.25)

d4(ψ(η)) := exp(

∫ r1

ψ(η)d2(s) ds

) 1

e2(ψ(η))> 0, (4.26)

D2 = D2(ξ, η) := d3(ξ)d4(ψ(η)), (4.27)

and

µ6 = d4(rs)h1(rs) < 0, µ7 = d4(rs)h2(rs) < 0, (4.28)

D3 = D3(ξ) :=

∫ ξ

r1

d3(s)h(s) ds ≥ 0, (4.29)

f2 = D2 · ♠ − (∂ηD2)e2w1 + d3(ξ)(

µ6 − d4(ψ(η)h1(ψ(η))))

(r∗ − rs)

+(

µ7 − d4(ψ(η))h2(ψ(η)))

∫ η

0w(ψ(s), s) ds. (4.30)

Note that

∂ηD2 = −D2ψ′(η)(

d2(ψ(η)) +e′2(ψ(η))

e2(ψ(η))

)

and ψ′(η) is small, one sees that f2 is still a higher order term. We then may write (4.24) as

∂ξ

(

D2w2 + µ6D3(ξ)(r∗ − rs) + µ7D3(ξ)

∫ η

0w(ψ(s), s) ds

)

−∂η(

D2e2w1

)

= f2. (4.31)


Equations (4.13)(4.31) are still not what we need for further investigations. There are in addition

two steps as followed to reformulate them. We first note thatD1(ψ(η), η) = 1,D2(ψ(η), η)e2(ψ(η)) =

1,D3(r1) = 0.

5. Normalization of the domains. To obtain the continuous dependence of solutions of problem

C(ψ∗) upon ψ∗, we introduce the following C3,α homeomorphism Θ : (ξ, η) 7→ (ξ, η) to normalize

the domain Ωψ to Ω := Ω0 = [0, 1]× [0, η0] without changing the periodical conditions on the upper

and lower boundaries:

Θ :

ξ =ξ − ψ(η)

r1 − ψ(η),

η = η,

or Θ−1 :

ξ = (r1 − ψ(η))ξ + ψ(η),

η = η.(4.32)

When |ψ| < r1 holds, we see that Θ−1 exists. It is straightforward to write (4.13)(4.31) in (ξ, η)

coordinates as

∂ξ(D1w1) + (r1 − rs)∂η(D1e1w2)

= (r1 − ψ)f1 + (ψ − rs)∂η(D1e1w2) − (ξ − 1)ψ′(η)∂ξ(D1e1w2), (4.33)

∂ξ

(

D2w2 + µ6D3(r∗ − rs) + µ7D3

∫ η

0w1(0, s) ds

)

− (r1 − rs)∂η(D2e2w1)

= (r1 − ψ)f2 − (ψ − rs)∂η(D2e2w1) + (ξ − 1)ψ′(η)∂ξ(D2e2w1). (4.34)

By Lemma 4.6 in [3], the norms of each function in (ξ, η) coordinates are equivalent to those in (ξ, η)

coordinates, and the equivalence does not depend on specific ψ∗. Thus the transformation intro-

duced here does not influence the estimates. Notice that in (4.33)(4.34) and the rest of the paper,

w1, w2 are considered as functions of (ξ, η), and Di = Di(ξ(ξ, η)), i = 1, 3;D2 = D2(ξ(ξ, η), η); ej =

ej(ξ(ξ, η)), j = 1, 2. So∫ η0 w(ψ(s), s) ds is transformed to

∫ η0 w1(0, s) ds as w1 = w1(ξ, η) in the

latter expression.

6. The problem now is that Di, ej (i = 1, 2, 3; j = 1, 2) are functions of ξ, η. We need to replace

them by functions depend only on the variable ξ for later convenience.

To this end, by setting µ8 = d4(rs) > 0 and

ej = ej(ξ) := (r1 − rs)ej((r1 − rs)ξ + rs) > 0, j = 1, 2, (4.35)

D1 = D1(ξ) := exp

∫ (r1−rs)ξ+rs

rs

d1(s) ds

> 0, (4.36)

D2 = D2(ξ) := exp

∫ (r1−rs)ξ+rs

r1

d2(s) ds

> 0, (4.37)

D3 = D3(ξ) :=

∫ (r1−rs)ξ+rs

r1

d3(s)h(s) ds ≥ 0, (4.38)


and let

f1 : = (r1 − ψ)f1 + (ψ − rs)∂η(D1e1w2) − (ξ − 1)ψ′(η)∂ξ(D1e1w2)

−∂ξ(

O(ψ − rs)D1w1

)

− ∂η(

O(ψ − rs)D1w2

)

, (4.39)

f2 : = (r1 − ψ)f2 − (ψ − rs)∂η(D2e2w1) + (ξ − 1)ψ′(η)∂ξ(D2e2w1)

−O(ψ − rs)(

∂ξ(D2w2) + µ6(r∗ − rs) + µ7

∫ η

0w1(0, s) ds

)

+∂η(

O(ψ − rs)D2w1

)

− µ7hD′3

η0η, (4.40)

equations (4.33)(4.34) are

∂ξ(

D1w1

)

+ ∂η(

D1e1w2

)

= f1, (4.41)

∂ξ

(

µ8D2w2 + µ6D3(r∗ − rs) + µ7D3

∫ η

0(w1(0, s) −

h

η0) ds

)

−∂η(

µ8D2e2w1

)

= f2. (4.42)

Here we have used boundary condition (4.9). These equations are finally what we need. Note that

fj = fj(U −U+b , ψ − rs) are all consist of higher order terms (with order at least to be two). Note

that f1 does not contain terms involving integrals as∫ η0 ·(0, s) ds.

4.3. Problem (NL). Now we formulate an equivalent form of problem (C(ψ∗)), where ψ∗ ∈ Sσ(see (3.11)). We call this as problem (NL(ψ∗)) or (NL) for simplicity. It consists of two parts.

1. Problem (NL1). This is a first order elliptic differential–integral system with nonlocal

boundary conditions:

(NL1) :

(4.41)(4.42)

Periodical conditions on η = 0, η = η0,

w2 = 0 on ξ = 1,

w2 + µ2

(

∫ η0 w1(0, s) ds − hη

η0

)

=(

g1 − µ2hηη0

)

+ µ3(r∗ − rs) on ξ = 0,

∫ η00 w1(0, s) ds = h on ξ = 0.

2. Problem (NL2). This is about two algebraic equations, i.e., Bernoulli’s law and invariance

of entropy along streamlines. For convenience, we denote the value of a function ϕ(ξ, η) on ξ = 0

as ϕ0. Then by (4.5), which is derived from Rankine–Hugoniot conditions, we see that

u0 − (u+b )0 = −µ5

(∫ η

0w1(0, s) ds− hη

η0

)

+

(

g2 −µ5hη

η0

)

+ µ4(r∗ − rs). (4.43)

Thus ρ0 and ρ, u may be determined by

(NL2) :

ρ0 =γ

γ − 1· p0

c0 − 12u

20(1 + w2

10),

ρ = ρ0

(

pp0

)1γ,

u =

21+w2 ·

(

c0 − γγ−1 · pρ

) 12.


Notice here that w10 := w1(0, η) should be solved from problem (NL1). So these two problems are

coupled.

5. Analysis of Linearized Problems

We will solve problem (NL) by finding a fixed point of a nonlinear mapping determined by the

linearized problem (L) of (NL) via BCMP. The construction of such a mapping and its properties

are closely connected with the solvability and estimates of problem (L). The new feature is that

there are certain nonclassical elliptic problems.

5.1. Linearized Problem (L). We first formulate the linearized problem (L) of the nonlinear

problem (NL).

1. Problem (L1). For given nonhomogeneous terms f1, f2 and g1, h, problem (L1) is to solve

the unknown functions w1, w2 and a number r∗ in the domain Ω = [0, 1] × [0, η0].

(L1) :

∂ξ(

D1w1

)

+ ∂η(

D1e1w2

)

= f1,

∂ξ

(

µ8D2w2 + µ6D3(r∗ − rs) + µ7D3

∫ η0 (w1(0, s) − h

η0) ds

)

−∂η(

µ8D2e2w1

)

= f2;

Periodical condition on η = 0, η = η0,

w2 = 0 on ξ = 1,

w2 + µ2

(

∫ η0 w1(0, s) ds − hη

η0

)

=(

g1 − µ2hηη0

)

+ µ3(r∗ − rs) on ξ = 0,

∫ η00 w1(0, s) ds = h on ξ = 0.

2. Problem (L2). Recall the convention we took in section 4.3 on ϕ0 := ϕ(0, η). For a given

nonhomogeneous term g2, problem (L2) is to compute u, ρ by

(L2) :

u0 − (u+b )0 = −µ5

(

∫ η0 w1(0, s) ds− hη

η0

)

+(

g2 − µ5hηη0

)

+ µ4(r∗ − rs),

p0 = (p+b )0 + (w2)0, p = p+

b + (w2),

ρ0 =γ

γ − 1· p0

c0 − 12 u

20(1 + (w1)

20),

ρ = ρ0

(

pp0

)1γ,

u =

21+(w1)2

·(

c0 − γγ−1 · pρ

) 12.

3. The following results concerning periodic functions are obvious.

Lemma 5.1. (1) Let f(η) be a periodic function with period T . Then

F (η) =

∫ η

0f(s) ds− η

T

∫ T

0f(s) ds

is also a periodic function with period T .

(2) Let φ(ξ, η) satisfy

∂ξφ = f1(ξ, η)

∂ηφ = f2(ξ, η)


in Ω := [0, 1] × [0, η0]. Then φ is periodic with respect to η with period η0 if and only if f1, f2 are

periodic with respect to η with period η0, and ∂ηf1 = ∂ξf2 holds in Ω, and∫ η00 f2(ξ, s) ds = 0 holds

for some ξ ∈ [0, 1].

For later convenience, we state the following assumption (P).

(P): We assume that the nonhomogeneous terms satisfy:

(1) f1, f2 ∈ C1,α(Ω), g1, g2 ∈ C2,α[0, η0], h ∈ R;

(2) f1, f2, g1 − µ2hη/η0, g2 − µ5hη/η0 are periodic with respect to η with period

η0.

5.2. Decomposition and Reformulation of Problem (L1). We see that problem (L1) is a

linear integral–differential elliptic system with boundary condition also involving integrals. Its

principle part is in divergence form, and the coefficients are smooth and depend only on the variable

ξ. We will write this problem as two nonlocal second order elliptic problems by utilizing linear

superposition and introducing certain “potentials”.

1. Since (L1) is linear, it is equivalent to the following two problems.

(L1

1) :

∂ξ(

D1w(1)1

)

+ ∂η(

D1e1w(1)2

)

= f1,

∂ξ

(

µ8D2w(1)2 + µ6D3(r

∗(1) − rs) + µ7D3

(

∫ η0 w

(1)1 (0, s) ds − hη

η0

))

−∂η(

µ8D2e2w(1)1

)

= 0;


w(1)2 = 0 on ξ = 1,

w(1)2 + µ2

(

∫ η0 w

(1)1 (0, s) ds− hη

η0

)

=(

g1 − µ2hηη0

)

+ µ3(r∗(1) − rs) on ξ = 0,

∫ η00 w

(1)1 (0, s) ds = h on ξ = 0.

(L2

1) :

∂ξ(

D1w(2)1

)

+ ∂η(

D1e1w(2)2

)

= 0,

∂ξ

(

µ8D2w(2)2 + +µ6D3r

∗(2) + µ7D3

∫ η0 w

(2)1 (0, s) ds

)

−∂η(

µ8D2e2w(2)1

)

= f2;


w(2)2 = 0 on ξ = 1,

w(2)2 + µ2

∫ η0 w

(2)1 (0, s) ds = µ3r

∗(2) on ξ = 0,∫ η00 w

(2)1 (0, s) ds = 0 on ξ = 0.

We see that

w1 = w(1)1 + w

(2)1 , w2 = w

(1)2 + w

(2)2 , r∗ = r∗(1) + r∗(2) (5.1)

are solutions to (L1).


2. Problem (L1

1). By the second equation in (L1

1) and Ω being simply connected, we may introduce

a potential function Φ(1) = Φ(1)(ξ, η) such that

w(1)2 =

1

µ8D2

(

∂ηΦ(1) − µ7D3

µ8e2(0)D2(0)

(∫ η

0∂ξΦ

(1)(0, s) ds − hµ8D2(0)e2(0)η

η0

)

−µ6D3(r∗(1) − rs)

)

, w(1)1 =

∂ξΦ(1)

µ8D2e2. (5.2)

It is important to notice here that the boundary condition w(1)2 = 0 on ξ = 1 is equivalent to

∂ηΦ(1)(1, η) = 0 since D3(1) = 0. So we may suppose Φ(1)(1, η) = m with m a number to be

determined. Furthermore, by Lemma 5.1, Φ(1) is periodical with respect to η with period η0 if w(1)i

(i = 1, 2) are, and the converse is also true.

Substituting (5.2) in the first equation of (L1

1), we get

∂η

D1e1µ8D2

∂ηΦ(1)(ξ, η) − µ6D1D3e1

µ8D2(r∗(1) − rs)

− µ7D1D3e1D2(0)e2(0)D2µ2

8

(∫ η

0∂ξΦ

(1)(0, s) ds − hµ8D2(0)e2(0)η

η0

)

+∂ξ

(

D1

µ8D2e2∂ξΦ

(1)(ξ, η)

)

= f1.

Let

a1(ξ) :=D1

µ8D2e2> 0, a2(ξ) :=

D1e1µ8D2

> 0,

a3(ξ) :=µ7D1D3e1

D2(0)e2(0)D2µ28

≤ 0;

µ9 :=µ2

e2(0)− µ7D3(0)

µ8D2(0)e2(0)> 0,

µ10 := µ3µ8D2(0) + µ6D3(0) < 0, µ13 := µ8e2(0)D2(0).

Then problem (L1

1) may be rewritten as

(L1′

1) :

∂ξ(

a1∂ξΦ(1)(ξ, η)

)

+ ∂η(

a2∂ηΦ(1)(ξ, η)

)

−a3

(

∂ξΦ(1)(0, η) − hµ13

η0

)

= f1,


Φ(1) = m on ξ = 1,

∂ηΦ(1) + µ9

(

∫ η0 ∂ξΦ

(1)(0, s) ds − µ13hηη0

)

= µ8D2(0)(g1 − µ2hηη0

) + µ10(r∗(1) − rs) on ξ = 0,

∫ η00 ∂ξΦ

(1)(0, s) ds = µ13h on ξ = 0.

Note that we have used D3(1) = 0 to simplify the expressions of boundary conditions on ξ = 1.

3. Problem (L2

1). Since Ω is a simply connected domain, we may introduce a potential Φ(2) =

Φ(2)(ξ, η) such that

w(2)1 =

−∂ηΦ(2)

D1, w

(2)2 =

∂ξΦ(2)

D1e1. (5.3)

It is easily seen that Φ(2) is periodic with respect to η. Substituting this in the rest equations of

(L2

1), we may write problem (L2

1) as


∂ξ

(

b1∂ξΦ(2)(ξ, η) + D3

(

µ6r∗(2) + µ7Φ

(2)(0, 0)))

+∂η(

b2∂ηΦ(2)(ξ, η)

)

− b3Φ(2)(0, η) = f2,


∂ξΦ(2) = 0 on ξ = 1,

∂ξΦ(2) − µ11Φ

(2) = D1(0)e1(0)(

µ3r∗(2) − µ2Φ

(2)(0, 0))

on ξ = 0.

where

b1(ξ) :=µ8D2

e1D1> 0, b2(ξ) :=

µ8D2e2D1

> 0, b3(ξ) := µ7D′3 > 0,

µ11 := (r1 − rs)e1(rs)µ2 > 0.

Here, by (4.6)(4.11) and D1(0) = 1, we know that µ11 > 0. Direct calculation shows that −µ2/µ3 =

µ7/µ6 = µ1. So we may take

r∗(2) = −µ1Φ(2)(0, 0), (5.4)

with Φ(2) the solution of

(L2′

1) :

∂ξ(

b1∂ξΦ(2)(ξ, η)

)

+ ∂η(

b2∂ηΦ(2)(ξ, η)

)

− b3Φ(2)(0, η) = f2,


∂ξΦ(2) = 0 on ξ = 1,

∂ξΦ(2) − µ11Φ

(2) = 0 on ξ = 0.

This is also a nonlocal elliptic problem. As far as we know, such problems have not been proposed

and studied before.

5.3. Solvability of Problem (L1′

1). Let Ψ = Ψ(ξ, η) := ∂ξΦ

(1)(ξ, η). Since a2 is a nonzero

function depending only on ξ, we can derive from (L1′

1) that Ψ satisfies the following problem

(ML1′

1) :

∂ξ

(

1a2∂ξ(

a1Ψ(ξ, η))

)

+ ∂η∂ηΨ(ξ, η) −(

a3a2

)′Ψ(0, η)

= ∂ξ

(

f1−a3hµ13/η0a2

)

,


∂ξ(a1Ψ) = f1 on ξ = 1,

∂ξ(a1Ψ) −(

a3 + µ9a2

)

Ψ

= f1 − a2µ8D2(0)∂ηg1 on ξ = 0.

Here we utilized a3(1) = 0 to obtain the boundary condition on ξ = 1, and an identity

µ2µ8a2D2(0) − a2µ9µ13 − a3µ13 = 0 on ξ = 0 (5.5)

to simplify the last boundary condition.

Lemma 5.2. Let (P) hold and Ψ be a solution to problem (ML1′

1). Then

∫ η0

0Ψ(0, s) ds = µ13h. (5.6)


Proof. Let

m(ξ) =

∫ η0

0a1(ξ)Ψ(ξ, s) ds.

Then by integrating the equation with respect to η from 0 to η0, we have

∂ξ

(

1

a2(ξ)m′(ξ) −

(a3

a2(ξ))m(0)

a1(0)− 1

a2(ξ)

∫ η0

0f1(ξ, s) ds

+hµ13a3

a2(ξ)

)

= 0.

By a3(1) = 0 and the boundary condition on ξ = 1:

m′(1) −∫ η0

0f1(1, s) ds = 0,

we hence get for ξ ∈ [0, 1],

m′(ξ) − a3(ξ)m(0)

a1(0)−∫ η0

0f1(ξ, s) ds+ hµ13a3(ξ) = 0. (5.7)

The boundary condition on ξ = 0 implies that

m′(0) − µ12m(0) =

∫ η0

0f1(0, s) ds − a2µ8D2(0)µ2h, (5.8)

where

µ12 := (a3(0) + µ9a2(0))/a1(0) > 0,

and we used that since g1 − µ2hη/η0 is periodical if assumption (P) holds, then∫ η0

0∂ηg1 ds = µ2h.

Taking ξ = 0 in (5.7) and by (5.8)(5.5), we get

m(0) =a2µ2µ8D2(0) − µ13a3(0)

µ12 − a3(0)a1(0)

· h = a1(0)µ13h.

This finishes the proof.

Once (ML1′

1) is solvable, we define the following equi-valued surface boundary value problem

(DL1′

1) :

1a2∂ξ(

a1∂ξϕ(ξ, η))

+ ∂η∂ηϕ(ξ, η) = a3a2

(

Ψ(0, η) − hµ13

η0

)

+ f1a2,


ϕ = m on ξ = 1,

ϕ = −µ9

∫ η0 (η − s)Ψ(0, s) ds + µ10(r

∗(1) − rs)η

+µ8D2(0)∫ η0 g1 ds+ hη2

2η0(µ9µ13 − µ2µ8D2(0)) on ξ = 0,

∫ η00 ∂ξϕ(0, s) ds = µ13h.

Where r∗(1) is determined by

r∗(1) − rs = − µ9

η0µ10

∫ η0

0sΨ(0, s) ds− µ8D2(0)

µ10η0

∫ η0

0g1 ds

+h

2µ10(µ9µ13 + µ2µ8D2(0)). (5.9)


Thus compatibility condition ϕ(0, 0) = ϕ(0, η0) holds. We note that r∗(1) indeed depends only on

f1, g1, h.

Lemma 5.3. If problem (ML1′

1) is solvable, then (ϕ, r∗(1)), with ϕ obtained from problem (DL1

′

1)

and r∗(1) determined by (5.9), is exactly a solution of problem (L1′

1).

Proof. We need only to show that

Ψ = ∂ξϕ.

To this end, set φ = Ψ − ∂ξϕ, then φ satisfies the following boundary value problem

∂ξ

(

1a2∂ξ(

a1φ(ξ, η))

)

+ ∂η∂ηφ(ξ, η) = 0,


∂ξ(

a1φ)

= 0 on ξ = 1,

∂ξ(a1φ) = 0 on ξ = 0,∫ η00 φ(0, s) ds = 0.

By maximum principle and Hopf boundary point lemma, we see a1φ, hence φ, should be zero.

Thus to solve the nonlocal elliptic boundary problem (L1′

1), we first show the unique solvability of

problem (ML1′

1). Then by setting r∗(1) as (5.9), the second step is to solve the equi-valued surface

problem (DL1′

1). We state the following result. The proof is presented in the next subsection.

Lemma 5.4. Let (P) hold. Then problem (ML1′

1) has a unique solution Ψ. The following estimate

also holds:

‖Ψ‖2+α ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.10)

Here and in the following ‖·‖k,α := ‖·‖k,α;Ω . By such a regularity of Ψ, we see Ψ(0, η) ∈ C2+α.

So the nonhomogeneous terms and r∗(1) in problem (DL1′

1) are well defined.

Lemma 5.5. Let (P) hold. Then problem (DL1′

1) has a unique solution ϕ. The following estimate

also holds:

‖ϕ‖3+α ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.11)

Proof. By Green theorem, (DL1′

1) is a typical equi-valued surface boundary problem with m to be

solved with the unknown ϕ together. By decomposition into Dirichlet problems, as in [19], it is

uniquely solvable and the estimate holds due to classical Schauder estimates for Dirichlet problems

[14].

Combining Lemma 5.3, Lemma 5.4 and Lemma 5.5, we have

Theorem 5.1. Let (P) hold. Then problem (L1′

1) has a unique solution (Φ(1), r∗(1)) and the fol-

lowing estimate holds:

|r∗(1) − rs| +∥

∥

∥Φ(1)

∥

∥

∥

3+α≤ C

(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.12)

5.4. Proof of Lemma 5.4. The proof of Lemma 5.4 is separated in the following several steps.


5.4.1. Uniqueness. Since a1(ξ) > 0 in [0, 1], we need only to show that

L1Ψ := ∂ξ

(

1a2∂ξΨ(ξ, η)

)

+ 1a1∂η∂ηΨ(ξ, η) − 1

a1

(

a3a2

)′Ψ(0, η) = 0,

Periodic conditions on η = 0, η = η0,

∂ξΨ = 0 on ξ = 1,

∂ξΨ − µ12Ψ = 0 on ξ = 0

(5.13)

has only the trivial solution Ψ ≡ 0.

Direct computation shows that

(

a3

a2

)′=

µ7

µ8e2(0)D2(0)D′

3(ξ) > 0.

(Note that h < 0 and µ7 < 0 hold by (4.21)(4.28).) Thus (1/a1)(a3/a2)′ > 0.

Since Ψ is periodic with respect to η, by Fourier transformation, we can express Ψ ∈ H1(Ω) as

Ψ(ξ, η) =∑

k∈Z

ψk(ξ)e√−1(

2kπη0

η)

.

Substituting this into (5.13), we get for k = 0,±1,±2, · · · ,

(

1a2ψ′k

)′− 1

a1

(

2kπη0

)2ψk − 1

a1

(

a3a2

)′ψk(0) = 0,

ψ′k(1) = 0,

ψ′k(0) − µ12ψk(0) = 0.

(5.14)

Lemma 5.6. Problem (5.14) has only the trivial solution ψk ≡ 0.

Proof. 1. Firstly we suppose that ψk is real-valued. If ψk(0) = 0, then since

1

a1

(

2kπ

η0

)2

≥ 0,

by maximum principle one gets ψk(ξ) ≡ 0.

2. If ψk(0) 6= 0, there will be contradictions. For example, if ψk(0) > 0, then

(

1

a2ψ′k

)′− 1

a1

(

2kπ

η0

)2

ψk > 0.

Thus by maximum principle, ψ can only attain its maximum at an end point. By Hopf boundary

point lemma, ξ = 1 is impossible to be a maximum point. But if ξ = 0 is a maximum point, Hopf

boundary point lemma implies that ψ′k < 0. Thus ψk(0) < 0 due to positiveness of µ12. This is a

contradiction.

The case ψk(0) < 0 can also be denied in a similar fashion.

3. Now if ψk is complex-valued, noting that a1, a2, a3, µ12 are all real, then by analyzing the real

and imaginary parts separately as above, it also yields ψk ≡ 0.

By the above lemma, we infer that Ψ ≡ 0 for (5.13).


5.4.2. Fredholm Alternative. Now we show the Fredholm alternative inH1(Ω) holds for the nonlocal

elliptic problem (ML1′

1), or equivalently,

L1Ψ = f ∈ (H1(Ω))∗,


∂ξΨ = g1 ∈ H−1/2[0, η0] on ξ = 1,

∂ξΨ − µ12Ψ = g0 ∈ H−1/2[0, η0] on ξ = 0.

(5.15)

Where (H1(Ω))∗ is the dual space of H1(Ω) and f, g1, g0 are periodic with respect to η with period

η0. We say that Ψ ∈ H1(Ω) is a weak solution of (5.15) if it is periodic with respect to η with

period η0 and for given f, g0, g1 as above, there holds

B[Ψ, v] : =

∫

Ω

1

a2∂ξΨ(ξ, η)∂ξv(ξ, η) +

1

a1∂ηΨ(ξ, η)∂ηv(ξ, η)

+1

a1

(

a3

a2

)′Ψ(0, η)v(ξ, η)

dξdη +µ12

a2(0)

∫ η0

0Ψ(0, s)v(0, s) ds

=1

a2(1)

∫ η0

0g1(s)v(1, s) ds − 1

a2(0)

∫ η0

0g0(s)v(0, s) ds + 〈f, v〉

for all v ∈ H1(Ω) which is periodic with respect to η with period η0. Note here that 〈·, ·〉 is the

pairing between (H1(Ω))∗ and H1(Ω).

Lemma 5.7. Precisely one of the following statements holds:

(1) For each f, g0, g1, problem (5.15) has exactly one weak solution u;

(2) There exists a nonzero weak solution u of (5.15) in the case f, g0, g1 are all zeros. In

addition, the dimension of the subspace in H1(Ω) which consists of such weak solutions u

is finite.

Proof. Let κ be a large positive number. Then

L1Ψ − κΨ = f ∈ (H1(Ω))∗,

Ψ = 0 on η = 0, η = η0,

∂ξΨ = g1 ∈ H−1/2[0, η0] on ξ = 1,

∂ξΨ − µ12Ψ = g0 ∈ H−1/2[0, η0] on ξ = 0

is uniquely solvable by trace theorem of Sobolev spaces and Lax-Milgram theorem [14], since κΨ

may control the nonlocal terms to obtain coerciveness. Then by classical methods for second order

elliptic equations (especially, compact embedding of H1(Ω) in L2(Ω)), one may obtain the Fredholm

alternative as in [14].

5.4.3. Regularity. By far we know that problem (ML1′

1), which is equivalent to

L1Ψ := ∂ξ

(

f1−a3hµ13/η0a2

)

,


∂ξΨ = f1 on ξ = 1,

∂ξΨ − µ12Ψ = f1 − a2µ8D2(0)∂ηg1 on ξ = 0,

(5.16)


has a unique solution Ψ ∈ H1(Ω) provided (P) holds, hence the requirements of Lemma 5.7 are

satisfied. That is, since a2 is smooth and bounded away from zero,

‖Ψ‖H1 ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.17)

Here and in the following we use ‖·‖Hk to be the norm of Sobolev space Hk(Ω), and ‖·‖∞ the norm

of L∞(Ω).

What left is to show that Ψ ∈ C2,α(Ω) and the estimate (5.10) holds.

1. L∞ estimate. Now we write (5.16) as

EΨ := ∂ξ

(

1a2∂ξΨ(ξ, η)

)

+ 1a1∂η∂ηΨ(ξ, η)

= 1a1

(

a3a2

)′Ψ(0, η) + ∂ξ

(

f1−a3hµ13/η0a2

)

,


∂ξΨ = f1 on ξ = 1,

∂ξΨ − µ12Ψ = f1 − a2µ8D2(0)∂ηg1 on ξ = 0.

(5.18)

Note that E is a second order elliptic operator, and this is a Robin problem. By trace theorem, we

see that Ψ(0, η) ∈ L2(Ω). Also note that f1(0, η), f1(1, η), ∂ηg1 ∈ H1(0, η0), so by the W 2,2 estimate

established in [14, 15], we have

‖Ψ‖W 2,2(Ω) ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

.

Then by embedding theorem,

‖Ψ‖∞ ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.19)

2. C2,α estimate. For problem (5.18), by classical Schauder estimate [14], we have

‖Ψ‖2+α ≤ C(

‖Ψ‖α +∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

≤ C(

ǫ ‖Ψ‖2+α +Cǫ ‖Ψ‖∞ +∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

.

Here we used the interpolation inequalities of Ck,α norms (see [14]). Then by (5.19), we get

‖Ψ‖2+α ≤ C(

∥

∥f1

∥

∥

1+α+ ‖g1‖2+α + |h|

)

.

This finishes the proof of Lemma 5.4.

5.5. Solvability of Problem (L2′

1). Problem (L2

′

1) is similar to problem (ML1

′

1). We have the

following theorem.

Theorem 5.2. Let f2 ∈ C1,α(Ω) be periodic with respect to η with period η0. Then problem (L2′

1)

has uniquely a solution Φ(2) ∈ C3,α(Ω), and the following estimate also holds:∥

∥

∥Φ(2)∥

∥

∥

3+α≤ C

∥

∥f2

∥

∥

1+α. (5.20)

Hence

|r∗(2)| ≤ C∥

∥f2

∥

∥

1+α. (5.21)

Proof. The proof is similar to that of Lemma 5.4.


5.6. Solvability of Problem (L1). Now we obtain the following important result.

Theorem 5.3. Let (P) hold. Then problem (L1) is uniquely solvable. In addition, the following

estimate holds:

|r∗ − rs| +2∑

k=1

‖wk‖2+α ≤ C

(

2∑

k=1

∥

∥fk∥

∥

1+α+ ‖g1‖2+α + |h|

)

. (5.22)

Proof. By (5.1)(5.2)(5.3) and Theorem 5.1, 5.2, this result is clear.

5.7. Solvability of Problem (L2). Once Theorem 5.3 holds, then we can easily solve u, ρ from

problem (L2).

Theorem 5.4. Let (P) hold and r∗, w1, w2 be obtained from problem (L1). Then problem (L2) is

uniquely solvable. In addition, the following estimate holds:

∥

∥u− u+b

∥

∥

2+α+∥

∥ρ− ρ+b

∥

∥

2+α≤ C ′

(

2∑

k=1

∥

∥fk∥

∥

1+α+

2∑

k=1

‖gk‖2+α + |h|)

. (5.23)

Proof. Note that the expressions in problem (L2) are all analytical functions. We remark that the

constant C ′ may depend on fk, gk, k = 1, 2, but if these nonhomogeneous terms are bounded, C ′ is

also bounded.

6. Proof of Theorem 3.5

Now we are in a position to prove Theorem 3.5 by Banach contraction mapping principle. We

first illustrate the ideas involved.

Let ψ∗(η) ∈ Sσ (see (3.11)), r ∈ Pκ (see (3.12)). Here κ = M0ε, σ = M1ε and M0,M1 are

constants depending only on the background solution, with M0 to be chosen later and M1 as

determined in the proof of Lemma 3.2. We define the following complete set Or,ψ∗

δ for any positive

number δ:

Or,ψ∗

δ =

U = (u,w, p, ρ)t ∈(

C2,α(Ω))4

:∥

∥U − U+b

∥

∥

2+α;Ω≤ δ,

U is periodic with respect to η with period η0

.

We emphasize here that for different r, ψ∗, the U+b may be different.

Now for a fixed (r, ψ∗) as above and U ∈ Or,ψ∗

δ , we set

ψ(η) = ψ∗(η) + r, (6.1)

fk = fk(U − U+b , ψ(η) − rs), k = 1, 2, (6.2)

gk = gk(U(ψ(η), η) − U+b (ψ(η), η),

U−(ψ(η), η) − U−b (ψ(η), η), ψ(η) − rs), k = 1, 2, (6.3)

h = h(U(ψ(η), η), U−(ψ(η), η)). (6.4)

Recall that the nonlinear terms fk, gk, h are defined by (4.39)(4.40), (4.8), (4.10) respectively. By

taking these as nonhomogeneous terms in problem (L), we can solve uniquely a pair (r∗, U ) with

U = (u, w = w1, p = w2 + p+b , ρ)

t. We denote the mapping U 7→ U by Nr,ψ∗ . We will show that

Nr,ψ∗ is a contractive mapping on Or,ψ∗

δ if ε (the perturbation of the supersonic flow) is small and


δ is chosen appropriately. That is, we can get uniquely one pair (r∗∗, U∗) with U∗ the fixed point

of Nr,ψ∗.

Next, we see that, by the above procedure, r 7→ r∗∗ also defines a mapping Nψ∗ : Pκ 7→ R. We

will show that this mapping is into and contracts on Pκ. We denote its unique fixed point by r∗ψ∗ .

Then by comparing the nonlinear problem (NL) with problem (L), we see that the pair (r∗ψ∗ , U∗ψ∗)

solves problem (NL), i.e., the “semi-fixed” boundary problem (C(ψ∗)), so the unique existence part

of Theorem 3.5 is proved. Here U∗ψ∗ is the fixed point of the mapping Nr∗

ψ∗ ,ψ∗ . We remark that the

procedure above is more complicated than that of [27], since the problem here is “semi-fixed”.

The continuous dependence part of Theorem 3.5 is proved by some estimates established later.

6.1. Construction of Nr,ψ∗. For simplicity of notations, we write Nr,ψ∗ as N and Or,ψ∗

δ as Oδ in

this subsection.

Proposition 6.1. N defined as above is a mapping from Oδ to itself if ε is small and δ is chosen

appropriately.

To prove this result, we need the following estimates of nonlinear terms.

Lemma 6.1. For fk, gk (k = 1, 2), h defined by (6.2)(6.3)(6.4), assumption (P) and the following

estimates hold:

‖gk‖2+α ≤ C(

δ2 + σ2 + κ2 + ε)

, (6.5)∥

∥fk∥

∥

1+α≤ C

(

δ2 + σ2 + κ2 + ε)

, (6.6)

|h| ≤ C(

δ2 + σ2 + ε)

. (6.7)

Here the constant C depends only on the background solution.

Proof. The verification is straightforward. To set the readers’ hearts at rest some computations are

provided here. Note that gk are defined in (ξ, η) coordinates, while some terms in fk are defined

in (ξ, η). But by Lemma 4.6 in [3], the resultant estimates of the same term in these different

coordinates are equivalent since the transformation Θ in (4.32) is a homeomorphism.

1. We first verify the assumption (P). Since f1 does not contain any terms like∫ η0 ·(s) ds, it

satisfies (P) since U , ψ∗ are periodical with respect to η with period η0. Due to the definition of h

and Lemma 5.1, (P) also holds for g1, g2 and f2, as can be checked by calculations.

For example, the terms in g1 involving integrals are

∫ η

0l(s) ds := −µ0

a+1

a+2 b

+1

∫ η

0

(

µ1w(ψ(s), s) − [uw]

[p]

∣

∣

∣

∣

U(ψ(s),s),U−(ψ(s),s)

ds

)

.

(The other terms are of course periodical.) So by definition of h and µ2,

∫ η0

0l(s) ds = −µ0

a+1

a+2 b

+1

µ1h = µ2h.

Thus Lemma 5.1 guarantees that g1 − µ2hη/η0 is periodical.

2. Now we demonstrate (6.5). Let us return back to (4.1). Note that G is analytical on U,U−.

Recall the definition of gkj (j = 1, 2; k = 1, 2, 3, 4) in section 4.1. By mean value theorem, we see


that, for example,

∥

∥g1j

∥

∥

2+α≤

∥

∥U(ψ(η), η) − U+b (ψ(η), η)

∥

∥

2+α

·∥

∥

∥

(

∂+G(U+b (ψ(η), η), U−

b (ψ(η), η)) − ∂+G(U+b (rs, η), U

−b (rs, η))

)∥

∥

∥

2+α

≤ Cδ ‖ψ(η) − rs‖2+α ≤ δ(κ + σ) ≤ C(δ2 + σ2 + κ2).

The terms g2j , g

3j , g

4j can be managed similarly. Such inequalities lead to (6.5).

Here and in the following we used the following fact [16]: ifm > 1, f, g ∈ Cm,α, then f g ∈ Cm,α

and

‖f g‖Cm,α ≤ C(

‖f‖Cm,α ‖g‖m+αC1 + ‖f‖C1 ‖g‖Cm+α + ‖f‖L∞

)

.

3. For (6.6), recalling f1 in (4.14), with λR there defined below (2.12), we have

‖f1‖1+α ≤ C(δ2 + σδ).

Next we turn to (4.39), the definition of f1. We see, for example,

∥

∥∂η(O(ψ − rs)D1w2)∥

∥

1+α≤

∥

∥O(ψ − rs)D1w2

∥

∥

2+α

≤ C(σ + κ)δ.

Thus (6.6) for k = 1 is clear.

For f2 (see (4.40)), the term ♠ (see (4.24)) appeared in f2 (see (4.30)) can only be controlled by

C(δ2 + σ2 + κ2 + ε) due to g3, which is bounded by C(δ2 + σ2 + κ2 + ε) as shown in 1. The rest

terms can be easily dominated by C(δ2 + σ2 + κ2). (6.7) is also clear from (4.10). Note that r∗ in

f2, f2 at this moment is replaced by r, the one chosen as in (6.1).

This completes the proof of this lemma.

Proof of Proposition 6.1. By Theorem 5.3 and 5.4, we see that (r∗, U) = N (r, U) satisfies

|r∗ − rs| +∥

∥U − U+b

∥

∥

2+α≤ C

(

δ2 + σ2 + κ2 + ε)

.

Note that the constant C depends only on the background solution. By choosing

δ = 2Cε, σ = C2ε, κ = 2Cε, (6.8)

ε ≤ ε0 ≤ min

1

12C2,

1

3M21

, (6.9)

where C2(= M1) has appeared in (3.21), we then have

|r∗ − rs| +∥

∥U − U+b

∥

∥

2+α≤ δ. (6.10)

Thus N is a mapping of Oδ to itself. By (6.8), we took M0 = 2C.

Remark 6.1. We emphasize here that the choice of M1 does not influence the constant C, which

depends only on the background solution (Lemma 6.1), but affects how small ε0 is. At this stage,

ε0 still can not be determined, but by boundary profile updating mapping, C2 = M1 depends solely

on C and background solution, thus C2 depends only on the background solution, as well as ε0

determined then.


6.2. Contraction of Nr,ψ∗ and its Continuous Dependence on r and ψ∗. For j = 1, 2, let

ψ∗j ∈ Sσ, r(j) ∈ Pκ, U (j) ∈ Or(j),ψ∗

j

δ . Set

ψ(j)(η) = ψ∗j (η) + r(j), (6.11)

f(j)k = fk(U

(j) − U+(j)b , ψ(j)(η) − rs), k = 1, 2, (6.12)

g(j)k = gk(U

(j)(ψ(j)(η), η) − U+b (ψ(j)(η), η), (6.13)

U−(ψ(j)(η), η) − U−b (ψ(j)(η), η), ψ(j)(η) − rs), k = 1, 2,

h(j) = h(U (j)(ψ(j)(η), η), U−(ψ(j)(η), η)). (6.14)

Here we emphasize again that U+b depends on ψ since for different ψ the definition domain of U+

b

is different.

Suppose that we get the pair (r∗(j), U (j)) from U (j) by the mapping Nr(j),ψ∗

j. In the following it

is sometimes convenient to write this as

(r∗(j), U (j)) = Nr(j),ψ∗

j(U (j)). (6.15)

We have the following proposition.

Proposition 6.2. If ε is small, then the following estimate holds:

|r∗(1) − r∗(2)| +∥

∥

∥(U (1) − U+(1)b ) − (U (2) − U

+(2)b )

∥

∥

∥

2+α(6.16)

≤ C3ε

(

|r(1) − r(2)| +∥

∥

∥(U (1) − U

+(1)b ) − (U (2) − U

+(2)b )

∥

∥

∥

2+α+ ‖ψ∗

1 − ψ∗2‖3+α

)

.

The constant C3 depends only on the background solution.

There are several implications of this estimate.

1. For ψ∗1 = ψ∗

2 = ψ∗, r(1) = r(2) = r, since U+(1)b = U

+(2)b , we may get

|r∗(1) − r∗(2)| +∥

∥

∥U (1) − U (2)

∥

∥

∥

2+α≤ C3ε

∥

∥

∥U (2) − U (2)

∥

∥

∥

2+α.

By choosing ε further small, this indicates that Nr,ψ∗ is contractive on Or,ψ∗

δ (recall (6.8) for

definition of δ). This shows that Nr,ψ∗ has exactly only one fix point (r∗ψ, Uψ) in Or,ψ∗

δ . This result

guarantees that the definition of the mapping Nψ∗ is reasonable.

2. By (6.10) we have |r∗ψ − rs| ≤ δ = 2Cε. Thus by (6.8) we see Nψ∗ : Pκ → Pκ.Next, we show the contraction of Nψ∗ . For the case that ψ∗

1 = ψ∗2 = ψ∗, but r(1), r(2) may be

different, we denote Nψ∗(r(i)) = r∗(i) for i = 1, 2. Then by definition of the mapping Nψ∗ , we have

(r∗(j), U (j)) = Nr(j),ψ∗

j(U (j)).

This means that now U (i) − U+(i)b = U (i) − U

+(i)b in (6.16). So from (6.16) we have contraction

|r∗(1) − r∗(2)| ≤ C3ε|r(1) − r(2)|

if C3ε ≤ 1/2.

This finishes the proof of unique existence part of Theorem 3.5.

3. Now for i = 1, 2, let ψ∗i ∈ Sσ and denote the fixed point of the mapping Nψ∗

iby r∗i , and the

fixed point of Nr∗i ,ψ∗

iby U∗

i . That is,

(r∗i , U∗i ) = Nr∗i ,ψ

∗

i(U∗

i ).


Then (6.16) yields

|r∗1 − r∗2| +∥

∥

∥(U∗1 − U

+(1)b ) − (U∗

2 − U+(2)b )

∥

∥

∥

2+α

≤ C3ε

1 −C3ε‖ψ∗

1 − ψ∗2‖3+α . (6.17)

Here, of course, U+(i)b is also obtained by applying the transformation Θψ(i) (see (4.32)) to the

restriction of U+b on Ωψ(i) (see (3.2) for the definition).

Now for ψi = r∗i +ψ∗i ∈ C3,α, by composition of functions we easily get the estimate (3.19) from

(6.17). This finishes the proof of continuous dependence part of Theorem 3.5.

So to prove Theorem 3.5, we need only to prove Proposition 6.2. To this end, we need the

following lemma, and it can be seen that Proposition 6.2 is an easy consequence of this lemma and

Theorem 5.3, Theorem 5.4.

Lemma 6.2. For ψ(j), f(j)k , g

(j)k , h(j) defined by (6.11)—(6.14), we have

2∑

k=1

∥

∥

∥f(1)k − f

(2)k

∥

∥

∥

1+α+

2∑

k=1

∥

∥

∥g(1)k − g

(2)k

∥

∥

∥

2+α+ |h(1) − h(2)| (6.18)

≤ Cε

(

|r(1) − r(2)| +∥

∥

∥(U (1) − U

+(1)b ) − (U (2) − U

+(2)b )

∥

∥

∥

2+α+ ‖ψ∗

1 − ψ∗2‖3+α

)

.

Proof. This lemma can be proved by direct computations as in the proof of Lemma 6.1, since all

the terms are at least of second order, and we know that κ, σ, U − U+b , U− − U−

b are all controlled

by ε now.

Acknowledgments

The authors would like to thank Prof. S. Chen for his interests, encouragements and many

advices on this work. H. Yuan’s research was partly supported by Shanghai Pujiang Program

05PJ14039. He also thanks Prof. X.-B. Pan for his constant encouragement and support.

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School of Mathematical Sciences, Fudan University, Shanghai 200433, China

E-mail address: [email protected]

Department of Mathematics, East China Normal University, Shanghai 200062, China

E-mail address: [email protected]

stability of cylindrical transonic shocks for two

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